The Orthogonal Universe

by Greg Egan

What would it be like to live in a universe with four dimensions that were all essentially the same?

The universe we inhabit has three dimensions of space and one of
time, and though relativity has taught us that there is no absolute
notion of time that is shared by everyone, the whole variety of
directions in space-time that different people might call “the future”
is entirely separate from the set of directions that different people
might call “north”.

What would be the outcome if that distinction were erased, and there
were four dimensions that were all as much alike as “north” and “east”?
Such a universe is the setting for a trilogy of novels that I’m
writing, with the overall title of Orthogonal. The first volume, The Clockwork Rocket, was published in 2011; the second, The Eternal Flame, has just been released by Night Shade books in the US, and will be out from Gollancz in the UK in October.

Since time as such is absent from the Orthogonal universe, a
first guess might be that it would resemble a snapshot of the world we
see around us at a single moment, albeit a snapshot with four dimensions
of space rather than three. Worse, it would be a snapshot with no
backstory: no sequence of prior events to organize and enrich the
subjects caught in the flash. It would consist of nothing but
scattered, isolated objects with no history or duration.

But it turns out that this guess would be wrong. What we see around us at a single moment takes the form it does because of
the existence of our own fourth dimension. It doesn’t make sense to
throw out that entire dimension, and then replace it by simply mimicking
what we originally saw in the remaining three.

So forget about snapshots. A more methodical approach would be to
take the equations that govern the behavior of matter and energy in our
universe, and make the smallest possible changes to them needed to put
all four dimensions on an equal footing. With the advent of relativity,
the equations of modern physics already come very close to treating
time and space even-handedly, often with the only difference being a plus sign appearing before a quantity involving distances and a minus sign
before a similar quantity involving times. If we rewrite the equations
slightly, turning those minus signs into plus signs, we can start to
make solid predictions about the nature of a universe where all the
dimensions are fundamentally the same.

Perhaps the most important thing to emerge is that, despite the
absence of a special time dimension, objects still end up with a kind of
persistence. In our own universe, we talk about the “world
line” of an object: the path it traces out in the four dimensions of
space-time as we follow its position over time. But even when we get
rid of time, the equations tells us that there will still be world
lines!

To see why this is true, the easiest place to start is to think about the kind of waves that we can expect the Orthogonal
universe to contain. In our universe, both light and matter have
fundamental wavelike properties, and it’s the geometry of these waves
that underlies the way objects move.

The simplest wave imaginable in the Orthogonal universe
consists of an endless series of equally spaced wavefronts, like an
idealized version of the waves you might get by shaking a long flat
board in calm water. In the figure below we only see two dimensions,
but that’s enough to show everything that matters, because as we move
along the other two dimensions nothing changes.

We’ve called one dimension “distance” and the other “time”, but as
far as the geometry is concerned there’s no difference: there’s nothing
special about the time dimension. We’ve also drawn a white arrow that
runs at right angles to all of the wave fronts, and gives us an
indication of the direction of the wave.

We’ll call the distance between wavefronts along the “distance” axis the wavelength of the waves, and the distance between wavefronts along the “time” axis the period of the waves.

Now, let’s compare this to another example, where we’ve taken exactly
the same kind of waves but changed their direction slightly with
respect to our “distance” and “time” axes.

The separation between these waves — measured directly from wavefront
to wavefront — is exactly the same as before. But because the wave’s
direction compared to our axes is different, the wavelength (measured
along the “distance” axis) has grown shorter, while the period (measured
along the “time” axis) has grown longer. We’ve drawn the waves with
the longer wavelength in red and the waves with the shorter wavelength
in violet, but the waves as such are actually identical in the two
diagrams; all that’s changed is their relationship to our chosen axes.

What happens if we combine several waves with slightly different directions? The result will look like this:

We see a series of points where the wavefronts intersect and
reinforce each other. These points all lie along a line that runs at
right angles to the wavefronts themselves — and this is how a
world line appears! If these waves described something like light, the
line where they reinforced each other would be the world line of a pulse
of light. If they were the quantum mechanical waves describing some
kind of matter, the line would be the world line of an elementary
particle.

Just as in our own universe, we can imagine successive
three-dimensional slices that intersect these world lines at slightly
different points for each slice, giving a picture of objects moving
about. So even without an official time dimension, we naturally end up
with concepts of change and motion that are very similar to those in our
own universe.

What’s more, all the simple geometry that we learned in school that
applies to distances in space now works just as well across all four
dimensions. Everyone knows that the shortest distance between two
points is a straight line, and that any detour or zig-zag only adds to
the distance. In a universe with four dimensions of space, what a
living creature would experience as “the passage of time” must really be
a form of distance, so the shortest time between two events must
also involve a straight line. Taking a detour when traveling from
event to event can only add to the time experienced by the traveler — in
stark contrast to the situation in our universe, where relativistic
time dilation means that less time passes for space travelers.

It’s that very simple twist that underlies the plot of my Orthogonal
trilogy. An alien civilization in the universe I’ve described are
facing a catastrophe that threatens to destroy their planet in a matter
of years, and their current technology is nowhere near sophisticated
enough to avert it. But by sending a group of travelers on a long
interstellar journey, those travelers will have a chance to spend
several generations trying to develop a solution, even though the
“straight line distance” to the catastrophe is just a few years.

Once the spacecraft ends up with its world line at right angles to
that of the home world (orthogonal to it, hence the title of the
trilogy), no time at all will be passing back home during that stage of
the journey, no matter how long it lasts for the travelers. What sets
the minimum time for the whole journey, as measured on the home world,
is the time it takes for the spacecraft to achieve that orthogonal state
— and as with a racing car trying to take a bend, the turning radius
depends on the force that is applied to make the trajectory change
direction.

The reversal of the usual kind of time dilation is only the beginning
of a long list of strange phenomena. The speed we measure for any
object or particle depends on the angle its world line makes with our
own, and since any world line is as good as another in the Orthogonal
universe, there can be no special speeds, such as the speed of light is
special for us. Light will be able to travel at any velocity
whatsoever, but the geometry of the waves means that the speed will be
tied to its wavelength, and hence its color.

Light of a shorter
wavelength — bluer light, in our own vocabulary — will travel faster
than longer-wavelength light. Just compare the first two wavefront
diagrams: when the direction of the wavefronts in the second diagram
makes a larger angle with the time direction, indicating a greater
velocity, the wavelength becomes shorter.

So when the aliens in this universe look up at the sky, every star’s
light will be spread out into a tiny spectrum, with the violet light
showing the star’s most recent position, while the red light shows the
star’s position centuries earlier — since the slower red light will have
taken longer to arrive.

It’s possible to map out the way this altered geometry leaves its
mark on everything, from the microscopic structure of matter to the
shape of the universe. More detail can be found on my web site, but if you don’t want to spoil too many surprises it’s best to read the novels first.

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I want to thank Greg Egan for this wonderfully comprehensive post and I will note that The Clockwork Rocket has been my top sf and sff title of 2011 for its combination of the sense of wonder you glimpsed above and wonderful characters you get to care about, however alien they are from some points of view.

I think this is addressed in the novel and possibly on the author's site where there is much more detail than in the necessarily condensed post above - you need almost infinite speed to go to the past; also there is the issue of "matter" vs "antimatter" which is hinted in the first book, but book 2 explores in depth

note that the "people" of the story are shape shifting beings made from very complicated molecules as the author speculates that those are the only "analog organic" such that can exist in the Orthogonal (Riemannian) Universe; actually pretty much any complex molecule has to be very complex

Anyway the author' site is the place to explore this fascinating universe and of course the novels that put a "human" face on it (despite the weirdish biology, the characters are generally compelling, though none from the second book quite reaches Yalda)

Frédéric, in principle there certainly is the possibility of travelling back in time exactly as you've described.

That raises a host of complications that are ultimately addressed in the third volume of the trilogy, The Arrows of Time, so I don't want to give spoilers for that book by discussing the issues in detail. At this point all I'll say is that the characters in the first book are still in the early stages of grappling with the physics, and are quite terrified enough of the prospect that, with the flight plan I've drawn here, they will be spending the second half of the journey with an arrow of time that runs backwards compared to the first half.

Mr. Egan, you are truly the most original speculative-fiction author out there, in my opinion. Your hands-on approach makes reading your books and website a treasure-trove of intellect-provoking gems. Can't wait to read this series! Thanks for sharing.-DB

It's a wonderfully clever idea. When I've taught Special Relativity, I've sometimes wondered what would happen if we changed that minus sign in the time equations so that all four dimensions took on the same form. As always, your ideas and writing are excellent.